Time-reversed nonlinear acoustics for wellbore integrity characterization

ABSTRACT

A pulsed sinusoidal acoustic signal transmitted through a subsurface volume of a wellbore may be detected. A time-reversed acoustic signal of the pulsed sinusoidal acoustic signal may be transmitted through the subsurface volume of the wellbore. Transmission of the time-reversed acoustic signal through the subsurface volume of the wellbore may result in generation of focused acoustic signal in the subsurface volume of the wellbore. The focused acoustic signal in the subsurface volume of the wellbore may be detected, and the integrity of the wellbore may be determined based on the focused acoustic signal in the subsurface volume of the wellbore.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation-in-part of U.S. patentapplication Ser. No. 16/083,441, filed on Sep. 7, 2018, which is anational stage application of International Application No.PCT/US2017/021606, filed 9 Mar. 2017, which claims the benefit of U.S.Provisional Patent Application No. 62/306,037 for “Time-ReversedNonlinear Acoustics For Downhole Pressure Measurements” by Paul A.Johnson et al., which was filed on 9 Mar. 2016, and of U.S. ProvisionalPatent Application No. 62/367,337 for “Time-Reversed Nonlinear AcousticsFor Downhole Pressure Measurements” by Paul A. Johnson et al., which wasfiled on 27 Jul. 2016, the entire contents of which Patent Applicationsare hereby specifically incorporated by reference herein for all thatthey disclose and teach.

STATEMENT REGARDING FEDERAL RIGHTS

The United States government has certain rights in this inventionpursuant to Contract No. 89233218CAN000001 between the United StatesDepartment of Energy and TRIAD National Security, LLC for the operationof Los Alamos National Laboratory.

PARTIES TO JOINT RESEARCH AGREEMENT

The research work described here was performed under a CooperativeResearch and Development Agreement (CRADA) between Los Alamos NationalLaboratory (LANL) and Chevron under the LANL-Chevron Alliance, CRADAnumber LA05C10518.

BACKGROUND OF THE INVENTION

Sedimentary rocks and man-made materials like concrete may be describedas a network of mesoscopic-sized “hard” elements (e.g., grains withcharacteristic lengths ranging from tens to hundreds of microns)embedded in a “soft” bond system (e.g., cement between grains, porespace, fluid). Such systems belong to a wider class of materialsreferred to as Nonlinear Mesoscopic Elastic Materials (NMEMs). Themicroscopic-sized imperfections at the interfaces between “hard” and“soft” subsystems are believed to be responsible for a number ofinteresting properties related to nonlinear and nonequilibrium dynamics,including the dependence of elastic parameters and attenuation on strainamplitude, slow dynamics, and hysteresis with end-point memory.Understanding and predicting these properties is basic for numerousapplications including oil exploration.

When high pressure zones are breached during drilling operations thehydrocarbon fluids travel up the well at a high rate, and drillingand/or extraction processes can be hampered and/or disrupted.

Knowledge of pore pressure in a formation is valuable for planningdrilling operations and for geochemical and geological analyses. Porepressures are the fluid pressures in the pore spaces in the porousformations. The pore pressure gradient is used in drilling fordetermining mud weight, which is selected based on pore pressuregradient, wellbore stability and fracture gradient prior to setting andcementing a casing. The drilling fluid is applied in the form of mudpressure to support the wellbore walls for preventing influx andwellbore collapse during drilling.

SUMMARY OF THE INVENTION

To achieve the purposes of embodiments of the present invention, asembodied and broadly described herein, the method for measuring porepressure in a formation, hereof, includes: generating a pulsedsinusoidal acoustic signal having a chosen frequency from a firsttransceiver disposed in a borehole in the formation; receiving theacoustic signal on at least one second transceiver disposed in theborehole; time reversing the received signal; transmitting thetime-reversed signals from the at least one second transceiver, wherebythe time-reversed acoustic signals form a focal volume centered on thefirst transceiver; receiving second and third harmonics of the chosenfrequency generated in the focal volume on the first transceiver, theharmonic signals having an amplitude; and monitoring the amplitude ofthe received harmonic signals.

In another aspect of embodiments of the present invention and inaccordance with its purposes the apparatus for measuring pore pressurein a formation, hereof, includes: a first signal generator forgenerating a pulsed sinusoidal signal having a chosen frequency; a firsttransceiver disposed in a borehole in the formation for receiving thesignal from the first signal generator, and transmitting an acousticsignal; at least one second transceiver disposed in the borehole forreceiving the transmitted acoustic signal and generating a firstelectrical signal therefrom; a processor for receiving the firstelectrical signal and time reversing the received signal; at least onesecond signal generator for receiving the time-reversed electricalsignal, generating a second acoustic signal therefrom, and directing thesecond acoustic signal onto the at least one second transceiver, suchthat the second acoustic signal is transmitted; whereby thetime-reversed acoustic signal forms a focal volume centered on the firsttransceiver, second and third harmonics of the chosen frequency aregenerated in the formation and received by said first transceiver,producing a second electrical signal having an amplitude, and theamplitude of the second electrical signal is monitored by the processor.

In yet another aspect of embodiments of the present invention and inaccordance with its purposes the method for measuring pore pressure in aformation through a borehole having a metal casing, hereof, includes:generating a pulsed sinusoidal acoustic signal having a chosen frequencyfrom a first transceiver disposed in the borehole; receiving acousticsignals on at least one second transceiver disposed in the boreholeabove the first transceiver; time reversing the received signals;transmitting the time-reversed signals with a selected intensity,whereby the time-reversed acoustic signals form a focal volume centeredon the first transceiver; receiving second harmonics of the chosenfrequency generated in the formation, on the first transceiver, thesecond harmonic signals having an amplitude; monitoring the amplitude ofthe received harmonic signals, whereby β is determined; receiving asecond acoustic signal responsive to vibrational excitation in the focalvolume on a third transceiver disposed in vibrational communication withthe metal casing; varying the intensity of the transmitted time-reversedacoustic signal; and measuring the time delay of the second acousticsignal relative to the time-reversed acoustic signal as a function ofthe intensity of the transmitted time-reversed acoustic signals; wherebya is determined.

In still another aspect of embodiments of the present invention and inaccordance with its purposes the apparatus for measuring pore pressurein a formation through a borehole having a metal casing, hereof,includes: a first signal generator for providing a pulsed sinusoidalsignal having a chosen frequency; a first transceiver disposed in theborehole in the formation for receiving the pulsed sinusoidal signalfrom the first signal generator, and transmitting an acoustic signal; atleast one second transceiver disposed in the borehole for receiving thetransmitted acoustic signal and generating a first electrical signaltherefrom; a first processor for receiving the first electrical signaland time reversing the received electrical signal; at least one secondsignal generator for receiving the time-reversed electrical signal,generating a second electrical signal therefrom, and directing thesecond electrical signal onto the at least one second transceiver, suchthat a second acoustic signal having a selected intensity istransmitted; whereby the time-reversed acoustic signals form a focalvolume centered on the first transceiver, the first receiver receivingsecond harmonics of the chosen frequency generated in the formation, theharmonic signals having an amplitude; a second processor for monitoringthe amplitude of the received harmonic signals, whereby β is determined;a third transceiver disposed in the borehole in vibrationalcommunication with the metal casing for measuring the time delay of thesecond acoustic signal relative to the time-reversed acoustic signal asa function of the selected intensity of the transmitted time-reversedacoustic signal; whereby a is determined.

Benefits and advantages of embodiments of the present invention include,but are not limited to, providing an apparatus and method fordetermining the existence of and the distance to a down holeover-pressured region in advance of a drilling bit, using a combinationof time reversal and elastic nonlinearity.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and form a part ofthe specification, illustrate the embodiments of the present inventionand, together with the description, serve to explain the principles ofthe invention. In the drawings:

FIGS. 1A and 1B provide graphs of the nonlinear parameters β and aα,respectively, as a function of pore pressure.

FIG. 2A is a schematic representation of an embodiment of the presentapparatus for practicing look-ahead drilling based on Time ReversalNonlinear Elastic Wave Spectroscopy, illustrating an acoustical sourcelocated above, and in the vicinity of a drill bit transmitting anacoustic wave that is swept in frequency as a function of time, thesignals propagating both into the well bore and into the surroundingformation, and received and recorded by transceivers disposed in thewellbore on a transceiver mount disposed near the drilling apparatus,while FIG. 2B illustrates the signals detected by the transceivers beingdigitally reversed-in-time by microprocessors located at thetransceivers or at the surface, amplified, and rebroadcast into the borehole and into the surrounding formation, and focusing the spacedistribution of elastic wave energy in the region of the originalacoustic source.

FIG. 3A is a schematic representation of an embodiment of thetransceiver mount illustrated in FIGS. 2A and 2B showing a possiblearrangement for the low-frequency transceivers, and FIGS. 3B and 3C showa possible arrangement for the mid-frequency and high-frequencytransceivers, respectively, while FIG. 3D illustrates a transceivermount having a transceiver at the focus of the other transceivers foruse in completed holes.

FIG. 4A illustrates the forward or calibration steps to achieve theclassical time reversed process using the apparatus of FIGS. 1A and 2B,and 3, while FIG. 4B illustrates reverse or focusing steps for the timereversed process.

FIG. 5 illustrates the ‘training’ process for reciprocal time reversal,where a predefined calibration signal is sequentially imposed ontransceivers, received by a transceiver at the focal position, andstored for use for time reversal processing once the stored signals aretime reversed.

FIG. 6 is a schematic representation of a laboratory apparatus forcarrying out reciprocal time reversal measurements for a simulated downhole environment.

FIG. 7A is a graph of the detected pulse propagation down the pipe inthe apparatus shown in FIG. 6, hereof, from a single transducer withouttime reversal, FIG. 7B is a graph of the detected pulse propagation fromall 10 transducers without time reversal, FIG. 7C is a graph of thedetected focused pulse propagation down the pipe from a singletransducer using classical time reversal, while FIG. 7D is a graph ofthe detected focused pulse propagation from all 10 transducers usingclassical time reversal.

FIG. 8A is a graph of the delay in the arrival time of a detected pulsein the pipe of the apparatus of FIG. 6 as the amplitude of the signalpulse increases from (a) to (c), while FIG. 8B is a graph of thedetected pulse as a function of frequency as the amplitude of the signalpulse increases from (a) to (c).

FIG. 9 is a graph of the pore pressure as a function of depth measuredby conventional techniques for an actual formation, permitting theprediction of gas/water contacts for locating hydrocarbon deposits.

FIG. 10 is a schematic representation of an embodiment of the apparatusused to determine α, and to verify that the determined value of α ismeaningful.

FIG. 11A is a graph of the time dependence of the particle velocity,while FIG. 11B is an expanded time period between 0.32 ms and 0.37 ms,both being measured at 500 mm from the source of vibrational excitationfor 20 source amplitudes.

FIG. 12A is a graph of the relative change in elasticity as a functionof strain estimated from the propagation of a pulse centered at 22.4 kHzand measured 500 mm from the source at 20 source amplitudes, while FIG.12B shows a sample applied pulse, the “star”, “circle” and “plus”symbols corresponding to the first five peaks the 9^(th)-21^(st) peak,and the final five peaks, respectively. peak, and the final five peaks,respectively.

FIG. 13A is a graph of the vibrational spectra measured on a sample ofBerea sandstone at 22 source amplitudes, the first 12 modes oflongitudinal vibration L₁ through L₁₂ being indicated, the star symbolsdenoting the location of the resonances for all source amplitudes, whileFIG. 13B clearly shows the shift in frequency for L₁₀ as an example.

FIG. 14 is a graph illustrating the relative frequency shift as afunction of the strain component ε_(xx) for the longitudinal modes L₄through L₁₂.

FIG. 15A illustrates an example probed volume.

FIG. 15B illustrates an example probed volume.

FIG. 15C illustrates an example probed volume.

FIG. 16 illustrates example changes in nonlinear response of materialsdue to micro annulus.

FIG. 17 illustrates example variations of nonlinearity near and awayfrom a fracture.

DETAILED DESCRIPTION OF THE INVENTION

As a result of the need for accurate pore pressure prediction fordrilling operators to reduce borehole trouble time and avoid drillingincidents, oil companies and oil service companies have been seekingmethods for detecting high pressures ahead drilling bits as theypenetrate the earth, such that corrective action can be taken before theregion is breached.

Overpressure rock has a signature elastic response that can be detectedby combining Time Reversal techniques with Elastic Nonlinearity in atechnique which is known as Time Reversal Nonlinear Elastic WaveSpectroscopy (TR NEWS). The nonlinear elastic wave response is directlyrelated to the effective pressure (hydrostatic load minus the porepressure). Time reversal is a method for focusing acoustic waves suchthat large wave amplitudes are obtained in a localized region of space.As a result of the large acoustic wave amplitudes at the focus and thenonlinearity of the material, harmonics may be generated (and sum anddifference frequencies if two waves are present). These harmonicfrequencies are detected at the focus and, as will be discussed in moredetail below, changes in the amplitude of the detected harmonicsindicate that high pressure may be present.

Nonlinear materials exhibit a nonlinear stress-strain relation which canbe probed by acoustic waves, leading to pressure-specific acousticsignatures. Harmonics of the incident acoustic frequencies are createdwhen the acoustic waves are focused. The effective pressure in aformation may be written as,P _(eff) =σ−bP   (1)where σ is the confining pressure, P is the pore pressure, and b is theBiot coefficient (typically 0.4-0.9 in rock). The effective pressure canalso be described by a nonlinear stress-strain relationship,

$\begin{matrix}{P_{eff} = {{{K\left\lbrack {1 - {\beta ɛ} - {\delta ɛ}^{2}} \right\rbrack}ɛ} + {K{\frac{\alpha}{2}\left\lbrack {{\left( {\left( {\Delta ɛ} \right)^{2} - ɛ^{2}} \right)\mspace{11mu}{sign}\mspace{11mu}\left( \overset{.}{ɛ} \right)} - {2\left( {\Delta ɛ} \right)ɛ}} \right\rbrack}}}} & (2)\end{matrix}$

where K is the linear stiffness constant, ε is the strain, Δε is thestrain amplitude, {dot over (ε)} denotes the partial derivative withrespect to time, sign is a function returning the sign (positive ornegative) of the argument, β and δ are combinations of third- andfourth-order elastic constants representing the acoustoelasticity(quadratic and cubic classical nonlinearity), and the parameter αrelates to the strength of the hysteresis, according to thePreisach-Mayergoyz model of elasticity. See, e.g., K. R. McCall et al.,“A new theoretical paradigm to describe hysteresis, discrete memory andnonlinear elastic wave propagation in rock,” Nonlin. Proc. Geophys. 3,89-101 (1996), R. A. Guyer et al., “Quantitative implementation ofPreisach-Mayergoyz space to find static and dynamic elastic moduli inrock,” J. Geophys. Res. 102(B3), 5281-5293 (1997), and G. DouglasMeegan, Jr. et al., “Observations of Nonlinear Elastic Wave Behavior inSandstone,” J. Acoust. Soc. Am. 94, (1993) 3387-3391. Combining Eqs. (1)and (2) leads to an expression of the pore pressure as a function ofconfining pressure and nonlinear elastic parameters of the material,

$\begin{matrix}{P = {\frac{1}{b}{\left( {\sigma - {{K\left\lbrack {1 - {\beta ɛ} - {\delta ɛ}^{2}} \right\rbrack}ɛ} + {K{\frac{\alpha}{2}\left\lbrack {{\left( {\left( {\Delta ɛ} \right)^{2} - ɛ^{2}} \right)\;{sign}\;\left( \overset{.}{ɛ} \right)} - {2\left( {\Delta ɛ} \right)ɛ}} \right\rbrack}}} \right).}}} & (3)\end{matrix}$The parameters α, β, and δ may be obtained from the time reversalsignal, with α being obtained from the velocity change of the focusedsignal as a function of strain amplitude. The velocity change may bealso measured using cross correlation or another standard technique on alow amplitude (linear) wave at the time reversal focus, and theprogressive delays caused by using progressively larger amplitudeexcitation waves. Cross correlation is a commonly applied method formeasuring time delays between a reference signal and a signal that hasexperienced a velocity change. β is obtained from the amplitudedependence of the second harmonic of a pulsed pure sinusoid or theamplitude dependence of sum (ω₁+ω₂) and difference (ω₁−ω₂) frequenciesif two waves are employed. See, also, TenCate, J. A. et al. (1996)“Laboratory Study of Linear and Nonlinear Elastic Pulse Propagation inSandstone,” J. Acoust. Soc. Am. 100(3), 1383-1391. δ is obtained fromthe amplitude dependence of the third harmonic of the fundamental driveamplitude at small, but still nonlinear amplitudes and, in general, canbe ignored. At larger amplitudes, however, a dominates and δ becomesoverwhelmed and can be ignored.

α is given by:

$\begin{matrix}{{\alpha = {\frac{\Delta C}{C_{0}}\frac{1}{ɛ}}}{{ɛ = \frac{\overset{¨}{u}}{2\pi fC_{0}}},}} & (4)\end{matrix}$where C₀ is the linear velocity and C the perturbed velocity. The secondderivative of u with respect to t is the particle acceleration measuredin the frequency domain, f is the wave fundamental frequency, and ε isthe strain measured at frequency f in the focal region as the signalsource amplitude is increased. By plotting the change in wave speed as afunction of strain, alpha can be obtained.

Alternatively, alpha can be obtained from the third harmonic amplitudealso when wave amplitudes are large. In the following alpha, beta anddelta are shown.

$\begin{matrix}{{\alpha = {\frac{c_{0}^{2}}{L}\frac{{\overset{¨}{u}}_{3f}}{{\overset{¨}{u}}_{1f}^{2}}}}{\beta = {\frac{c_{0}^{2}}{L}\frac{{\overset{¨}{u}}_{2f}}{{\overset{¨}{u}}_{1f}^{2}}}}{\delta = {\frac{\omega c_{0}^{3}}{L}\frac{{\overset{¨}{u}}_{3f}}{{\overset{¨}{u}}_{1f}^{3}}}}} & (5)\end{matrix}$where L is the wavelength of the fundamental frequency divided by two,equivalent to the radius of the focal region, the second derivative of uwith respect to time, 3f, is the third harmonic acceleration amplitude,the second derivative of u with respect to time, 2f, is the secondharmonic acceleration amplitude, the second derivative of u with respectto time, 1f, is the fundamental harmonic acceleration amplitude, andω=2πf, where f is the fundamental frequency.

FIG. 1A is a graph of β as a function of pore pressure, while FIG. 1B isa graph of α as a function of pore pressure.

Time reversal permits the generation of focused, intense (non-damaging)sound in a region to induce local nonlinearities if high pressure ispresent, by taking advantage of the above relation for u_(2f), therebypermitting detection and imaging of overpressure regions. As an example,waves may be introduced into a specimen using a piezoelectrictransducer. The waves are recorded on another transducer locatedelsewhere on the sample surface. The recorded waves are then reversed intime, and emitted from the detecting transducers, where they followtheir forward wave paths backwards-in-space, and coalesce, focusing atthe original source transducer, since the elastic wave equation issymmetric with respect to time. That is, the wave equation may beevaluated either forward or backward in time, the physics beingidentical. Amplitudes at the time-reversed focus are large due toconservation of energy, since all of the energy contained in thelong-duration scattered-signal is collapsed onto the focal point inspace and time. Since wave amplitudes are largest at the focus, thelocal response may be nonlinear, but only at the focus.

Further, by measuring α and β for a formation using time reversaltechniques, one can obtain accurate values for the pore pressure in aformation, using Equations 2 and 3, above. Among the uses for thegradient of the pore pressure are the prediction of gas/water contacts,which permit more accurate location of hydrocarbons in the formation.

Reference will now be made in detail to the present embodiments of theinvention, examples of which are illustrated in the accompanyingdrawings. In the FIGURES, similar structure will be identified usingidentical reference characters. Turning now to FIG. 2A, a schematicrepresentation of an embodiment of the present apparatus, 10, forpracticing look-ahead drilling based on TR NEWS is illustrated.Acoustical source, 12, located above, and in the vicinity of drillingbit, 14, turned by shaft, 15, transmits a sine wave that is swept infrequency as a function of time. Other signal types may be employed,including pulsed signals. Signals, 16, propagate both into well bore,18, and into formation, 20, surrounding the borehole, and are receivedand recorded by transceivers, 22, also disposed in the wellbore ontransceiver mount, 24, disposed near the drilling apparatus. One end oftransceiver mount 24 may be placed between just behind drilling bit 14and 10-20 m therefrom. Apparatus, 25, is employed to change thedirection of drilling bit 14 in response to pore pressure measurementsin accordance with embodiments of the present invention. As shown inFIG. 2B, the detected signals are digitally reversed-in-time, bymicroprocessors located at detectors (transceivers) 22 or at thesurface, amplified (not shown in FIGS. 2A and 2B), and rebroadcast, 26,into the bore hole and into the surrounding formation. In accordancewith the time-reversal process, the signals follow their forwardpropagation directions in reverse, and focus at source 12 that also actsas a detector. The phase relationships among the returning waves permitthe constructive interference thereof resulting in space and timefocusing. The focused signal is large in amplitude and is effective forinducing an elastic nonlinear response in focal volume, 28. If a portionof focal volume 28 encompasses a high fluid or gas pressured region, thenonlinear response of generated harmonic frequencies (and potentiallysum and difference frequencies), and time delays due to wave speeddecreases, will be significantly greater than at an established baselinethereof. This nonlinear response is detected and interpreted at thesurface or by microprocessors located behind the drilling string (notshown in FIGS. 2A and 2B). By varying the frequency of the swept sinesignal, the distance to the over-pressured region can be estimated fromthe frequency, if the sound velocity in formation 20 is known (as isgenerally the situation) using the relationship for the wavelength,λ=velocity/frequency. The diameter of the focal spot measured at thehalf maximum value is equal to one-half of the dominant wavelength. See,e.g., “Depth Profile of A Time-Reversal Focus in An Elastic Solid,” byMarcel C. Remillieux et al., Ultrasonics 58 (2015) 60-66. Correctiveaction can then be taken by the drillers (placing blow-out stops etc.).

A schematic representation of an embodiment of transceiver mount 24 isshown in FIGS. 3A-3C. FIG. 3A shows a possible arrangement forlow-frequency transceivers, 22 a, while FIGS. 3B and 3C show anarrangement for mid-frequency, 22 b, and high-frequency transceivers, 22c, respectively. The transducer mount includes a long metallic portion,30, adapted to fit in a cased borehole having an inner diameter of 6 in.Transducer mount 24 may also be constructed of sturdy plastics capableof withstanding high temperatures and caustic environments.Advantageously, the long dimension of mount 24 is equal to or largerthan five times the largest wavelength of the elastic waves propagatingin the formation. If mount 24 is made of steel (Young modulus E=200 GPaand mass density ρ=8500 kg/m³), and is operated at a center frequencyf_(c)=2 kHz, then the elastic wavelength is λ˜√(E/ρ)/f_(c)=2.6 m, andthe minimum length of the mount would be about 12 m. The actual shape ofthe mount can be optimized to improve the transfer of energy from thetool to the formation.

FIG. 3D is a schematic representation of another embodiment oftransceiver mount 24 where transceiver 12 is located approximately inthe center of the mount. This design is suitable for measuring porepressure in formations through well casings.

Independently controlled low-frequency transceivers 22 a mid-frequencytransceivers 22 b and high-frequency transceivers 22 c controlled bydigital synthesizers, 32, 34, and 36, respectively, which are directedby microcontroller and digital signal processor, 38, are affixed alongmount 24 to provide the required excitation signals. Transducers vary insize and relative spacing depending on the center frequency ofexcitation signal that is intended to be generated. For low frequencyexcitation, large transducers are distributed over the entire length ofthe tool. For high frequency excitation, smaller transducers arecentered with a smaller span around the point where focus should beachieved (at transceiver 12).

Source 12 generates a swept sine wave that encompasses frequenciesf_(i . . . l) that provide the spatial resolution λ_(i . . . l) ofinterest in a given group of strata. For example, given a typicalformation velocity c of 2000 m/s, and a desired probe distance of l=10 min advance of the drill bit, the time-reversed focal diameter would bed=20 m, and the center frequency would therefore be f_(j)=100 Hz. Usinga swept sine wave f_(i . . . 1), spatial wavelengths above and belowthis value may be probed. The spatial wavelength may be reduced byincreasing the frequency until the large nonlinear response disappears.In this manner the distance to the over-pressured region can bedetermined.

FIG. 4A illustrates the forward or calibration steps to achieve theclassical time reversed process using the apparatus of FIGS. 2A and 2B,and 3, while FIG. 4B illustrates that reverse or focusing steps for thetime reversed process. In Step 40, of FIG. 4A, a predeterminedcalibration signal, P, from memory storage is directed to an arbitrarywaveform generator, which drives, Step 42, source transducer 12 at thefocal position, S. Signals generated from source transducer 12 arereceived by all of the transceivers 22, Step 44, and stored in memory,Step 46. Step 48 in FIG. 4B simultaneously directs the time reversedsignals stored in memory in step 46 to an arbitrary waveform generatorwhich drives transceivers 22, Step 50, the transmitted signals beingfocused onto transducer 12 in Step 52. Nonlinear signals, F, generated,Step 54, in the focal area are stored in memory for later processing.This calibration process would be undertaken every time measurements ofα and β are to be made, since external conditions, such as increasingpressures of drilling mud, may change the calibration.

Noise from impulsive elastic waves generated from the action of drillingbit 14 on the materials in a formation can be used as a source for theclassical time reversal measurements in place of acoustic source 12 inaccordance with embodiments of the present invention. In this situation,the drilling bit would be stopped when the amplified time-reversedsignals generated by transceivers 22 are employed to generate harmonicsin front of drilling bit 14, the harmonic signals being correlated withthe time-reversed signals from the drilling bit.

The method described in FIG. 4 is based on ‘classical’ time reversal. Aswill now be described, ‘reciprocal’ time reversal can also be applied.Here, transceivers 22 located on transceiver mount 24 are caused toindividually broadcast a signal for a ‘training’ step. Transceiver 12detects these signals that are broadcast one at a time. The detectedsignals are time reversed, and amplified and reemitted from transceivers22. They again focus on the detector 12. This process works due to thereciprocity of the wave equation, since the transfer function in onedirection is the same as that in the other direction. FIG. 5 illustratesthe ‘training’ process. A predefined calibration signal, P, in memorystorage is directed, Step 56, to an arbitrary waveform generator, whichdrives transceivers 22, Step 58, sequentially, the generated signalsbeing focused onto transceiver 12, Step 60, and the received signalsstored in memory for time reversal processing, Step 62, after eachstored signal is time reversed. See, e.g., T. J. Ulrich et al., “Timereversal and non-linear elastic wave spectroscopy (TR NEWS) techniques,”Intl. J. Non-Linear Mech. 43 (2008) 209-216.

Having generally described embodiments of the present invention, thefollowing EXAMPLES provides additional details.

EXAMPLE 1

FIG. 6 is a schematic representation of an apparatus for carrying outreciprocal time reversal measurements described above in the laboratoryfor a simulated down hole environment. Pipe, 100, was embedded in block,102, of Berea Sandstone. Ten acoustic transceivers 22 were affixed toportion, 104, of pipe 100 emerging from block 102. A reference signal,for example, a pulsed (10-20 kHz) sinusoidal waveform having a 50 kHzbandwidth, is directed into at least one arbitrary waveform generator,106, by computer 38. After amplification by at least one poweramplifier, 108, each generator signal is directed to a singletransceiver 22, one generator signal at a time. The signal travelingthrough pipe 100 and block 102 is recorded by laser vibrometer, 110,after being received by fiber optic attachment, 112, disposed insidepipe 100. The signal received for each emission is digitized bydigitizer, 114, and directed to computer 38, which time reverses each ofthe received signals and programs the arbitrary waveform generators 106with the time-reversed signals. Reciprocal time reversal, in its basicform, includes first sending the last digitized element, then the secondto last digitized element and so forth until the entire waveform hasbeen inverted. Focusing in pipe 100 may be improved (typically, tosharpen the focus), by filtering the time-reversed signal (not shown inFIG. 6). Signal averaging may also be performed on the detected signals.The final step is for all of the arbitrary waveform generator tosimultaneously direct the time-reversed signals to the transducersthrough the power amplifiers. During that phase, all the generators aresynchronized via pulser, 116. While all the transceivers are emitting,laser vibrometer 110 records the signals generated in the focal volumethat are digitized and analyzed for nonlinear components by computer 38.

The time-reversed signals may be broadcasted successively at differentamplitudes to assist in the detection of the nonlinear signals. Asdiscussed above, the size of the region probed by focused waves in theformation depends of the wavelength used for the first reference signal.

FIG. 7A is a graph of the detected pulse propagation down pipe 100 froma single transducer 22 without time reversal, while FIG. 7B is a graphof the detected pulse propagation from all 10 transducers 22 withouttime reversal. FIG. 7C is a graph of the detected focused pulsepropagation down pipe 100 from a single transducer 22 using reciprocaltime reversal, while FIG. 7D is a graph of the detected focused pulsepropagation down pipe 100 from all 10 transducers using classical timereversal.

The signal strength increases by a factor of 10 when using reciprocaltime reversal over that resulting from the use of conventional sources.This is clear example of an apparatus capable of transmitting elasticwave energy to a formation in a simulated borehole/casing/rock systemusing the method in accordance with embodiments of the presentinvention.

FIG. 8A is a graph of the delay in the arrival time of a detected pulsein the pipe of the apparatus of FIG. 6 as the amplitude of the signalpulse increases from (a) to (c). As mentioned above, this shift isrelated to α. FIG. 8B is a graph of the detected pulse as a function offrequency as the amplitude of the signal pulse increases from (a) to(c). The fundamental as well as the second and third harmonics arereadily observable. As discussed above, β may be obtained by monitoringthe second harmonic. Monitoring the third harmonic is an alternativemethod for obtaining α. When the bandwidth of the fundamental isbroader, however, the third harmonic can partially overlap with thesecond harmonic, which may make third harmonic measurements moredifficult. Additionally, the third harmonic may be influenced by δ (seeEquations 5 above). Due to the relative sizes of α and δ, this latterissue is generally not a serious problem, but having an alternative wayof measuring α, such as by using the time delay, is advantageous.

FIG. 9 is a graph of the pore pressure as a function of depth measuredby conventional techniques for an actual formation, permitting theprediction of gas/water contacts (GWC) for locating hydrocarbondeposits.

Once these contacts are located, drilling can be redirected usingapparatus 25 in FIG. 2A to accomplish the change in direction.

EXAMPLE 2

Propagation of an Impulsive Elastic Waveform in a Long, Thin Bar

As discussed above, with the aid of time reversal, elastic wave energyis focused at a point in space and an impulsive waveform will begenerated. Since this process involves waves traveling throughmaterials, and material properties may be strain dependent, the arrivaltime of the impulsive waveform may be dependent on the amplitude of theexcitation. The term of hysteretic nonlinearity α in the equation ofstate (Equ. 3) governs this effect.

To verify α can be quantified by monitoring the propagation speed of anelastic wave as a function of the strain amplitude, laboratoryexperiments were performed. Although the propagation of impulsiveelastic waves remains the principal measurement, time reversal is notrequired to generate the strain since the measurements are restricted toa one-dimensional waveguide over a known propagation distance. Thehysteretic nonlinearity parameter has never been measured in thismanner, so the determination is validated using nonlinear resonantultrasound spectroscopy.

FIG. 10 is a schematic representation of the apparatus, 118, used todetermine α in sample, 120, of Berea sandstone (Cleveland Quarries,Amherst, Ohio) having a length of 1794 mm (70.63 in) and a diameter of39.6 mm (1.56 in). The sample was supported by a foam pad, not shown inFIG. 10, in order to simulate free (unconstrained) boundary conditions.Elastic waves were generated using a piezoelectric transducer, 122,epoxied onto one flat end of sample 120. Impulsive elastic waveformsgenerated in signal generator, 124, and amplified by voltage amplifier,126, were propagated in sample 120 at different amplitudes. Thevibrational response of bar 120 was recorded on the surface of thesample using 3D Laser Doppler vibrometer, 128, and received by dataacquisition apparatus, 130.

Returning to FIGS. 2A, 2B, and 6, if one places a vibrational motionsensor, such as a piezoelectric transducer, 132, on wall, 134 ofwellbore, 18, in FIGS. 2A and 2B, corresponding to pipe 100 in FIG. 6,one can measure the particle velocity as a function of excitationintensity of the waves generated in the formation by time reversal, fromwhich the peak strain (Equ. 8) can be evaluated in focal volume 28. Asdescribed in this EXAMPLE 2 for laboratory experiments, from the timehistory of the particle velocity and the peak strain, α can beevaluated. With β being available from measurement of the intensity ofharmonics, the pore pressure can be determined. Note that since themetal walls or casings of wellbores exhibit linear response tovibrations, the particle velocity, which is the motion excited in thefocal region is accurately measured through the wall.

FIGS. 11A and 11B show the time histories of the axial component of theparticle velocity measured at 500 mm from the source 122 for 20 sourceamplitudes ranging from 1 to 20 Vpp in steps of 1 Vpp (beforeamplification), with FIG. 11B illustrating an expanded abscissa between0.32 ms and 0.37 ms. The original sinusoidal waveform observed at thelowest amplitudes progressively evolves into a triangular wave, as aresult of hysteretic nonlinearity, with additional distortion caused byclassical nonlinearity. It is observed that the waveforms experience asignificant delay in arrival time as the source amplitude increases.This delay is observed not only at the extrema (peaks of amplitude) butalso at the zero crossings. Classical nonlinearity would not induce anydelay at the zero crossings (where the strain is equal to zero), becauseit produces an instantaneous variation of the modulus without timeconstants involved. Therefore, the observed delay is a directconsequence of hysteretic nonlinearity.

The data shown in FIGS. 11A and 11B can be further reduced to quantifyhysteretic nonlinearity. For a signal measured at the source amplitudei, the signal measured at the source amplitude i-1 is taken as areference to compute a relative time delay, Δt^(i/i-1)/t₀. The relativetime delay between two signals is estimated by cross-correlation. Thetime delay between the lowest source amplitude (i=0) and the sourceamplitude i is then obtained by summation as,

$\begin{matrix}{{\Delta t^{i/0}\text{/}t_{0}} = {\left( {\sum\limits_{n - 1}^{i}{\Delta t^{{n/n} - 1}}} \right)\text{/}t_{0}}} & (6)\end{matrix}$The relative time delay between the signals is also equal to therelative change in speed of the longitudinal wave, Δc^(i/0)/c₀. Further,at the perturbation level, the relative change in the Young's modulus E(the modulus involved in the propagation of a longitudinal wave in along thin bar) is related to the relative change in the speed of thelongitudinal wave as,ΔE ^(i/0) /E ₀=2Δc ^(i/0) /c ₀   (7)The relative changes in the elastic modulus over the propagation path ofthe waveform can be followed as a function of the maximum strainamplitude at the measurement point. The strain component of interest isε_(xx), where x is axial direction. The strain component ε_(xx) can beexpressed analytically as a function of the axial component of theparticle velocity v_(x) as,

$\begin{matrix}{ɛ_{xx} = \frac{v_{x}}{c}} & (8)\end{matrix}$Recall that the particle velocity is obtained from the vibrationalmotion measured by 3D Scanning Laser Doppler Vibrometer 110.

Reduced data from FIGS. 11A and 11B are shown in FIGS. 12A and 12B.Three distinct regimes may be identified in the waveform: (i) theinitial portion of the waveform, where the dynamic strain at themeasurement point transitions from zero to steady state, and labeledwith a “star”; (ii) the center portion of the waveform, where theabsolute value of the peak amplitude of the dynamic strain isquasi-constant (that is, steady state), and labeled with a “0”; and(iii) the final portion of the waveform, where the dynamic strain at themeasurement point transitions from steady state back to zero, that is, apath similar to the earlier portion of the waveform but in reverse, andalso labeled with a “+”. In the initial portion of the waveform, theelastic modulus decreases progressively from its undisturbed equilibriumvalue to a smaller value (ultimately up to 5.3% smaller at the largestsource amplitude employed as the steady state is approached). This isthe portion typically referred to as conditioning. In the center potionof the waveform, the evolution of the material softening with the strainamplitude is essentially independent of the wave cycle selected for theanalysis (that is, similar results are obtained from the 9^(th) to21^(st) peaks of the waveforms). In this regime, the slope of the curveis 10480±600 (dimensionless). This value quantifies the hystereticnonlinearity, but it will be determined in EXAMPLE 3 below how thisvalue compares to the parameter α.

EXAMPLE 3

Nonlinear Resonant Ultrasound Spectroscopy:

Returning to FIG. 10, with minor modifications, the apparatus may beutilized for resonant ultrasound spectroscopy. Sample, 120, of Bereasandstone (Cleveland Quarries, Amherst, Ohio) having a length of 1794 mm(70.63 in) and a diameter of 39.6 mm (1.56 in) is shown. The sample wassupported by a foam pad, not shown in FIG. 10. Piezoelectric transducer122 was driven with a sequence of harmonic voltage signals. Eachharmonic signal was applied for 55 ms, and the transient vibrationalresponse was recorded during the last 40 ms of the source signal, toensure that steady state conditions had been reached. Vibrationalspectra were constructed from the harmonic responses. Frequenciesranging between 0.3 kHz and 7 kHz in steps of 2.5 Hz were employed, andat 22 excitation amplitudes ranging between 0.25 to 10 Vpp (beforeamplification). The axial component of the acceleration was measured onthe flat end opposite to the source by an accelerometer usingtransducer, 132, the output of which was processed by signalconditioner, 134, and analyzed using data acquisition apparatus 136. Theresonance frequencies are not computed for the first three modes becauseof the poor signal-to-noise ratio.

Any resonance mode can be selected to quantify hysteretic nonlinearityas long as the mode type is purely longitudinal. The vibrational spectrafor this experiment are shown in FIGS. 13A and 13B. Material softening,a drop in the effective elastic moduli resulting from the fact thatelastic moduli are dependent on the strain amplitude and strain rate, isobserved when the drive amplitude of the source becomes sufficientlylarge (Equ. 3). For a given drive amplitude, such softening can bequantified by plotting the relative frequency shift as a function of themaximum strain in the sample, the slope of which is the nonlinearparameter α. The maximum strain in the sample may be inferredanalytically from the measured vibrational response. For thelongitudinal modes, the strain component of interest is ε_(xx). For asystem having a one-dimensional geometry and unconstrained boundaries,the expression given in Eq. (8) can also be used in the context of aresonance experiment to relate the maximum amplitude of the axialcomponent of the particle velocity at the free end of the sample (wheredata is acquired) to the maximum amplitude of the axial component of thestrain in the sample.

As depicted in FIG. 14, the relative frequency shift varies almostlinearly with the maximum strain beyond 4 microstrains. When a mode hassufficient data beyond this strain value, the points may be linearlyfitted and the slope of this fit calculated. It appears that materialsoftening converges to a single value (all curves superimpose) ofα=5260±160 for resonance modes L₆ through L₁₂. Below the 6^(th)resonance mode, the elastic response does not have a sufficiently largeamplitude to reach the threshold strain value of about 4 microstrain.

The slope of the relative change of the resonance frequency isapproximately twice the value of the relative change of the Young'smodulus observed in the pulse propagation experiment, which isconsistent with the analytical relationship between Young's modulus andresonance frequency of a longitudinal mode at the perturbation level,ΔE/E ₀=2Δf/f ₀   (9)Therefore, the quantification of hysteretic nonlinearity in the pulsepropagation experiment and with nonlinear resonant ultrasoundspectroscopy are equivalent.

In the pulse propagation experiments, the Young's modulus isapproximately constant below 4 microstrains in the conditioning phaseand varies linearly with strain above this value, with a sharptransition between the two regimes (see FIGS. 11A and 11B). In theresonance experiment, the resonance frequency also varies linearly withstrain beyond 4 microstrains but experiences a smooth transition to thisregime. During this transition, classical nonlinearity plays asubstantial role. Classical nonlinearity can induce a frequency shift inthe resonance experiment, but cannot induce a time delay in the pulsepropagation experiment; hence, the sharp transitions in FIGS. 11A and11B. In this sense, it is possible to decouple the contributions fromclassical nonlinearity and nonequilibrium dynamics in the pulsepropagation experiment.

In summary, application of a method that combines time reversal andelastic nonlinearity (TR NEWS) provides the means to quantitativelyprobe for over pressured regions in advance of the drilling bit, and todetermine the distance to an over pressured region. Moreover, gas/watercontacts may be located in accordance with the teachings of the presentinvention, and drilling directed to more successfully locatehydrocarbons.

Time-reversal technique disclosed herein may be used to determinecharacteristics of materials, such as subsurface materials. Subsurfacematerials may refer to materials located beneath the surface/locatedunderground. Subsurface materials may refer to materials that are notexposed at the surface of the ground. Characteristics of materials mayrefer to attributes, features, qualities, properties, and/or othercharacteristics of materials. Time-reversal technique disclosed hereinmay be used to determine qualitative and/or quantitative characteristicsof materials. The present disclosure may be used to provide a highlyfocused and discrete measurement method using nonlinear acoustics andTime Reversed (TR) source propagation at downhole conditions.

For instance, time-reversal technique (e.g., classical time-reversaltechnique, reciprocal time-reversal technique) may be used to determineintegrity of subsurface structures, such as a wellbore. Time-reversaltechnique may be used to focus wave energy in a subsurface volume ofinterest to probe/explore nonlinear properties of materials within thesubsurface volume of interest, and the integrity of the wellbore may bedetermined from the nonlinear properties of materials. Exploration ofother subsurface materials are contemplated.

For example, one or more transceivers (e.g., a first transceiver) may beconfigured to transmit one or more pulsed sinusoidal acoustic signalsthrough a subsurface volume of interest, such as a subsurface volume ofa wellbore. A pulsed sinusoidal acoustic signal may be transmittedthrough entirety of the subsurface volume or through a part of thesubsurface volume. For example, the transceiver(s) may be located at thecenter of the subsurface volume and the pulsed sinusoidal acousticsignal may be transmitted outward from the center of the subsurfacevolume to one or more transceiver located outside the subsurface volume.

The subsurface volume of the wellbore include the region being probedusing the time-reversal technique. The subsurface volume of the wellboremay include one or more materials are that to be probed using thetime-reversal technique. For example, the subsurface volume of thewellbore may include casing, rock, and concrete between the casing andthe rock, and/or other materials.

A pulsed sinusoidal acoustic signal may have a particularfrequency/wavelength. That is, the transceiver(s) may transmit, throughthe subsurface volume, a pulsed sinusoidal acoustic signal of aparticular frequency/wavelength. In some implementations, thefrequency/wavelength of the pulsed sinusoidal acoustic signal maydetermine the size of the subsurface volume of interest (e.g., thesubsurface volume of the wellbore). That is, the frequency/wavelength ofthe pulsed sinusoidal acoustic signal may determine the size of theregion being probed using the time-reversal technique. The size of thesubsurface volume being probed may be determined by the wavelength λ(inverse of the frequency) of the pulsed sinusoidal acoustic signal. Forexample, the subsurface volume may be a hemisphere with (approximate)radius of λ/4 to λ/2. Thus, the frequency/wavelength of the pulsedsinusoidal acoustic signal may determine how large of a volume is probedusing the time-reversal technique. The subsurface volume may extendvertically and/or laterally. For instance, the frequency/wavelength ofthe pulsed sinusoidal acoustic signal may determine the depth of theregion being probed. By changing the frequency/wavelength of the pulsedsinusoidal acoustic signal, the size (e.g., depth) of the volume beingprobed may be changed.

The time-reversal technique may utilize an integrated effect to probedifferent locations. For example, a particular frequency of the pulsedsinusoidal acoustic signal may be used to probe a subsurface volume thatpenetrates to a certain depth in the ground and determinecharacteristics of materials into that depth. The frequency of thepulsed sinusoidal acoustic signal may be changed to increase the depthof penetration and probe deeper into the ground. Characteristics ofmaterials added in the deeper penetration may be determined by takinginto account the characteristics of the materials in the original probe.For instance, the original probe may penetrate into a casing todetermine the characteristics of the casing. The deeper probe maypenetrate into both the casing and concrete adjacent to the casing. Themeasurements from the deeper probe may be deconvolved to effects fromthe casing and effects from the concrete, and the characteristics of theconcrete within the deeper probe may be determined. Thus, the size ofthe subsurface volume that is being probed may be changed (e.g.,incrementally increased) to determine characteristics of subsurfacematerials in different locations.

One or more transceivers (e.g., a second transceiver) may be configuredto detect the pulsed sinusoidal acoustic signal(s) transmitted thoughthe subsurface volume of interest (e.g., the subsurface volumepenetrated by the wellbore). Detecting a pulsed sinusoidal acousticsignal may include one or more of identifying, measuring, receiving,and/or otherwise detecting the pulsed sinusoidal acoustic signal. Thepulsed sinusoidal signal(s) may be time-reversed for transmission backthrough the subsurface volume of interest. The transceiver(s) thatdetected the pulsed sinusoidal acoustic signal(s) and/or othertransceiver(s) may be configured to transmit the time-reversed acousticsignal(s) of the pulsed sinusoidal acoustic signal(s) through thesubsurface volume of interest. Thus, the time-reversed acousticsignal(s) may be transmitted back through the subsurface volume beingprobed.

The time-reversal technique may utilize time-reversed acoustic signal(s)to probe the subsurface volume of interest. The Green function thatdictates propagation may cause the time-reversed acoustic signal(s) tocreate a version, or an approximation, of the original source pulse. Thetime-reversal technique may utilize reciprocity (e.g., reciprocity ofwave equation in the Green function) to focus wave energy in thesubsurface volume of interest. For example, transmission of thetime-reversed acoustic signal(s) through the subsurface volume of thewellbore may result in generation of focused acoustic signal(s) in thesubsurface volume of the wellbore. The focused acoustic signal(s) mayfocus wave energy in the subsurface volume of the wellbore. The focusedacoustics signal(s)/focused wave energy may be used to probe thecharacteristics of the materials within the subsurface volume of thewellbore.

One or more transceivers inside and/or near the subsurface volume ofinterest (e.g., the subsurface volume of the wellbore) may be configuredto detect the focused acoustic signal in the subsurface volume of thewellbore. Detecting the focused acoustic signal may include one or moreof identifying, measuring, receiving, and/or otherwise detecting thefocused acoustic signal. For example, the transceiver(s) thattransmitted the pulsed sinusoidal acoustic signal(s) through thesubsurface volume of interest and/or one or more transceivers coupled tomaterials being probed may be used to detect the focused acousticsignal. The characteristics of materials within the subsurface volume ofinterest may be determined based on the focused acoustic signal and/orother information. For example, the integrity of the wellbore may bedetermined based on the focused acoustic signal in the subsurface volumeof the wellbore and/or other information. The integrity of the wellboremay include and/or be affected by the integrity of the casing, integrityof the rock around the casing, the integrity of the concrete between thecasing and rock, the integrity of the casing-concrete interface, theintegrity of the concrete-rock interface, and/or the integrity of othermaterials inside/around the wellbore

In some implementations, the characteristics of materials that aredetermined based on the focused acoustic signal may include one or morevalue(s) of one or more nonlinear material parameters. A nonlinearmaterial parameter may refer to a parameter of material that exhibits anonlinear response to one or more stimuli. A nonlinear materialparameter may refer to a parameter of material that exhibits nonlinearresponse to the focused acoustic signal/focused wave energy. Thetime-reversal technique may utilize time-reversed acoustic signal(s) tofocus energy into the subsurface volume of interest and measure thenonlinear material parameter(s) in the subsurface volume of interest.

In some implementations, the nonlinear material parameter(s) that aremeasured using focused acoustic signal/focused wave energy may includehigher-order elastic moduli and/or other nonlinear material parameters,such as alpha, beta and delta. The focused acoustic signal and/or otherinformation may be used to determine the values of nonlinear materialparameters alpha, beta, delta, and/or other nonlinear materialparameters in the subsurface volume of interest. The values of nonlinearmaterial parameters alpha, beta, delta, and/or other nonlinear materialproperties may change with changes in materials within the subsurfacevolume of interest. For example, the values of nonlinear materialparameters alpha, beta, delta, and/or other nonlinear materialproperties may change with changes in integrity of the wellbore, such aswith formation of defect in the wellbore. The values of nonlinearmaterial parameters alpha, beta, delta, and/or other nonlinear materialparameters may be used to determine other physical properties ofmaterials in the subsurface volume of interest.

The value(s) of nonlinear material parameter(s) in the subsurface volumeof interest may be used to determine other characteristics of materialswithin the subsurface volume of interest. For example, the integrity ofthe wellbore may be determined based on the value(s) of nonlinearmaterial parameter(s) in the subsurface volume of the wellbore and/orother information. That is, determination of the integrity of thewellbore based on the focused acoustic signal in the subsurface volumeof the wellbore may include determination of one or more values of oneor more nonlinear material parameters in the subsurface volume of thewellbore based on the focused acoustic signal, and determination of theintegrity of the wellbore based on the value(s) of the nonlinearmaterial parameter(s) in the subsurface volume of the wellbore.

In some implementations, different portions of the wellbore/surroundingmaterial may be explored by changing the size (e.g., depth) of thesurface volume of the wellbore (by changing the frequency of the pulsedsinusoidal acoustic signal). This provides the ability to probe materialcharacteristics at specific locations (e.g., specific distance fromborehole scan). For example, the frequency of the pulsed sinusoidalacoustic signal may be set to different values to inspect the casing inthe wellbore, the rock around the casing, the concrete between thecasing and rock, the casing-concrete interface, the concrete-rockinterface, and/or other locations inside/around the wellbore.

For instance, FIGS. 15A, 15B, and 15C illustrate example sizes of probedvolume 1550, 1560, 1570 using different frequency of the pulsedsinusoidal acoustic signal. Use of different frequency enables differentportions of the wellbore/surrounding region to be probed. For example,the frequency of the pulsed sinusoidal acoustic signal may be set toprobe the casing 1500 in FIG. 15A. The frequency of the pulsedsinusoidal acoustic signal may be set to probe the casing 1500, thecasing-concrete interface 1510, and the concrete 1520 and in FIG. 15B.The frequency of the pulsed sinusoidal acoustic signal may be set toprobe the casing 1500, the casing-concrete interface 1510, the concrete1520, the concrete-rock interface 1530, and the rock 1540 in FIG. 15C.Other sizes and locations of probed volume are contemplated.

While the probed volume are shown as being spheres in FIGS. 15A, 15B,and 15C, this is merely as examples and is not meant to be limiting. Forexample, the actual volume that is probed may depend on the propertiesof materials involved, which may distort the shape to a non-idealizedsphere/shape. For instance, the wave speed in the fluid in the boreholemay be less than in the concrete, rock, or steel layers. This may resultin the wavelength in the borehole being smaller for the same frequency,causing the probed volume to be smaller in the borehole. On the otherhand, the steel layer may have the highest wave speed of all the layersand may thus have the largest wavelength, resulting in increase in thevolume of probing. The formation and concrete layers may have wavespeeds between these extremes, which may impact the shape/size of thevolume of probing. Other factors that impact the shape/size of thevolume of probing may include transceiver radiation pattern, placementof the transceiver (e.g., centralized in the borehole or pushed to oneside), coupling to specific wave types (e.g., Stonely waves), and/orborehole modes.

In some implementations, characteristics of materials within thesubsurface volume of interest may be determined based on comparison ofthe value(s) of the nonlinear material parameter(s) with baselinevalue(s) of the nonlinear material parameter(s). For example, thedetermination of the integrity of the wellbore based on the value(s) ofthe nonlinear material parameter(s) may include determination of theintegrity of the wellbore based on comparison of the value(s) of thenonlinear material parameter(s) with baseline value(s) of the nonlinearmaterial parameter(s). For instance, the integrity of the wellbore maybe determined based on comparison of measured values of nonlinearmaterial parameters alpha, beta, delta, and/or other nonlinear materialparameters in the subsurface volume of the wellbore with baseline valuesof nonlinear material parameters alpha, beta, delta, and/or othernonlinear material parameters.

A baseline value of a nonlinear material parameter may refer to a valueof the nonlinear material parameters that is used for comparison orcontrol. A baseline value of a nonlinear material parameters may includeinitially/previously measured value of the nonlinear material parameter.For example, a baseline value of a nonlinear material parameter mayrefer to a value of the nonlinear material parameters that was measured(using the time-reversal technique) when one or more characteristics ofmaterial in the subsurface volume of interest is known. For example, abaseline value of a nonlinear material parameter may refer to a value ofthe nonlinear material parameters that was measured when the integrityof the wellbore is known (e.g., after completion of the wellbore).

Changes in the value(s) of the nonlinear material parameter(s) from thebaseline value(s) may indicate changes in characteristics of material inthe subsurface volume of interest. For example, deviation of thevalue(s) of the nonlinear material parameter(s) from the baselinevalue(s) of the nonlinear material parameter(s) (different between thevalue(s) and the baseline value(s)) may indicate formation of one ormore defect in the subsurface volume of the wellbore. For instance, thesubsurface volume of the wellbore may include casing, rock, and concretebetween the casing and the rock, and/or other materials, and thedeviation of the value(s) of the nonlinear material parameter(s) fromthe baseline value(s) of the nonlinear material parameter(s) mayindicate formation of one or more defects in the casing, one or moredefects in the rock, one or more defects in the concrete, one or moredefects at casing-concrete interface, one or more defects atrock-concrete interface, and/or other defect between the times at whichthe baseline value(s) and the value(s) were measured. For example,formation of micro-cracks and/or macro-cracks in the subsurface volumeof the wellbore may lead to higher nonlinear response in the subsurfacevolume of the wellbore.

In some implementations, extent of the deviation of the value(s) of thenonlinear material parameter(s) from the baseline value(s) of thenonlinear material parameter(s) may indicate extent of the defect(s) inthe subsurface volume of the wellbore. For example, the amount ofdifference between the value(s) and the baseline value(s) may indicatean amount of defect(s) that have formed in the subsurface volume of thewellbore between the measurements. For instance, increase in values ofnonlinear material parameters alpha, beta, and/or delta may indicateformation of defect(s) in the subsurface volume of the wellbore, withlarger values of nonlinear material parameters alpha, beta, and/or deltacorresponding to larger defect(s) in the subsurface volume of thewellbore. In some implementations, the value(s) of nonlinear materialparameters alpha, beta, and/or delta may decrease based on the defectdestroying the elastic nature of the materials. For example, if thedefect(s) get very large, the defect(s) may destroy the elasticproperties of the materials, which may result in low value(s) ofnonlinear material parameters alpha, beta, and/or delta. In someimplementations, the comparison of the values and baseline value(s) mayinclude tracking how the value(s) changes from the baseline value(s).That is, rather than simply looking at whether and by how much thevalue(s) have changed from the baseline value(s) at a moment in time,the profile of changes in the value(s) from the baseline value(s) may beused to determine formation/evolution of defects.

FIG. 16 illustrates example changes in nonlinear response of materialsdue to micro annulus. FIG. 16 shows example relative frequency shift asa function of excitation amplitude for three different casings. Thethree casings may have different sizes of micro annulus between thecasings and the cement around the casings. The value of nonlinearmaterial parameter alpha may be calculated from the slopes of curvesshown in FIG. 16. As shown in FIG. 16, the curve with the largest slope(largest value of alpha) corresponds to the casing with the largestmicro annulus while the curve with the smallest slope (smallest value ofalpha) corresponds to the casing with the no micro annulus.

FIG. 17 illustrates example variations of nonlinearity near and awayfrom a fracture. FIG. 17 shows example variation of nonlinearitymeasured using time reversal on an open face formation, near and awayfrom the fracture. The delay of arrival of the peak signal indicates awave speed decrease (or modulus softening) and may be proportional tothe nonlinear material parameter alpha. From the changes in the measuredsignal (e.g., change in size, wavelength, delay of arrival) as afunction of the amplitude of vibration, value of nonlinear materialparameter alpha may be calculated. Value of nonlinear material parameteralpha may be calculated from the slope of the lines. As shown in FIG.17, nonlinearity of material increases closer to the fracture. Baselinenonlinear response may provide baseline shift, and the measured shiftedmay be compared with the baseline shift to determine formation offracture in the material (more shifting when there is fracture/closer tothe fracture).

In some implementations, characteristics of materials within thesubsurface volume of interest may be determined based on the value(s) ofthe nonlinear material parameter(s) without measuring baseline value(s)of the nonlinear material parameter(s). Rather than separately measuringbaseline values for materials in the subsurface volume of interest,information regarding the materials may be used to interpret themeasured value(s) of the nonlinear material parameter(s). For example,information may be available on types and/or configuration of casing,rock, and/or concrete being probed. Such information may be used todetermine the value(s) of nonlinear material parameter(s) that would beexpected for the materials in different condition (e.g., value(s)expected without defect, value(s) expected with defect, value(s)expected with different types of defect, value(s) expected with defectin different locations). With knowledge on the type of casing used inthe wellbore, concrete used in the wellbore, rock surrounding thewellbore, a single measurement of the value(s) of the nonlinear materialparameter(s) may be taken to determine (estimate) the characteristics ofmaterials (e.g., defect) within the subsurface volume of the wellbore.

In some implementations, the types of defects in the subsurface volumeof interest may be determined based on the value(s) of the nonlinearmaterial parameter(s) and/or other information. For example,determination of the integrity of the wellbore based on the value(s) ofthe nonlinear material parameter(s) may include determination of type(s)of defect in the subsurface volume of the wellbore based on the value(s)of the nonlinear material parameter(s). The value(s) of the nonlinearmaterial parameter(s) may be used to distinguish between different typesof defects in the subsurface volume of the wellbore. For example,different value(s) of the nonlinear material parameter(s) may correspondto different types of defect (e.g., microcracks, delamination, weakeningof matrix, fractures) in different types of material (e.g., casing,concrete, rock, casing-concrete interface, concrete-rock interface), andthe type of the defect may be determined based on the value(s) of thenonlinear material parameter(s) (e.g., specific value(s) of thenonlinear material parameter(s), certain amount of change between thevalue(s) and the baseline value(s), certain evolution of change from thebaseline value(s) to the value(s).

In some implementations, the location of defects in the subsurfacevolume of interest may be determined based on the value(s) of thenonlinear material parameter(s) and/or other information. For example,determination of the integrity of the wellbore based on the value(s) ofthe nonlinear material parameter(s) may include determination oflocation(s) of defect in the subsurface volume of the wellbore based onthe value(s) of the nonlinear material parameter(s). The value(s) of thenonlinear material parameter(s) may be used to perform spatial mappingof the subsurface volume of the wellbore to find locations of defects(e.g., weaker points in space according to the value(s) of the nonlinearmaterial parameter(s))

The foregoing description of the invention has been presented forpurposes of illustration and description and is not intended to beexhaustive or to limit the invention to the precise form disclosed, andobviously many modifications and variations are possible in light of theabove teaching. The embodiments were chosen and described in order tobest explain the principles of the invention and its practicalapplication to thereby enable others skilled in the art to best utilizethe invention in various embodiments and with various modifications asare suited to the particular use contemplated. It is intended that thescope of the invention be defined by the claims appended hereto.

What is claimed is:
 1. A method for determining wellbore integrity, the method comprising: transmitting a pulsed sinusoidal acoustic signal through a subsurface volume of a wellbore, the pulsed sinusoidal acoustic signal having a frequency; detecting the pulsed sinusoidal acoustic signal transmitted though the subsurface volume of the wellbore; transmitting, through the subsurface volume of the wellbore, a time-reversed acoustic signal of the pulsed sinusoidal acoustic signal, transmission of the time-reversed acoustic signal through the subsurface volume resulting in generation of focused acoustic signal in the subsurface volume of the wellbore; detecting the focused acoustic signal in the subsurface volume of the wellbore; and determining the integrity of the wellbore based on the focused acoustic signal in the subsurface volume of the wellbore, wherein the determining the integrity of the wellbore based on the focused acoustic signal in the subsurface volume of the wellbore includes determining values of nonlinear material parameters alpha, beta, and delta in the subsurface volume of the wellbore based on the focused acoustic signal, and comparing the values of the nonlinear material parameters alpha, beta, and delta with baseline values of the nonlinear material parameters alpha, beta, and delta, wherein the baseline values of the nonlinear material parameters alpha, beta, and delta are previously measured values of the nonlinear material parameters alpha, beta, and delta with known integrity of the wellbore, further wherein the nonlinear material parameter alpha relates to a strength of hysteresis and the nonlinear material parameters beta and delta relate to acoustoelasticity.
 2. The method of claim 1, wherein the frequency of the pulsed sinusoidal acoustic signal determines a size of the subsurface volume of the wellbore that is probed, and the integrity of the wellbore is determined by increasing the size of the subsurface volume of the wellbore that is probed via one or more changes in the frequency of the pulsed sinusoidal acoustic signal, further wherein characteristics of subsurface materials determined by an initial probing of the subsurface volume of the wellbore with a smaller size of the probing is accounted for in a later probing of the subsurface volume of the wellbore with a larger size of the probing to determine characteristics of subsurface materials included in the larger size of the probing and not included in the smaller size of the probing.
 3. The method of claim 1, wherein the nonlinear material parameters alpha, beta, and delta include higher-order elastic moduli.
 4. The method of claim 1, where deviation of the values of the nonlinear material parameters alpha, beta, and delta from the baseline values of the nonlinear material parameters alpha, beta, and delta indicates formation of a defect in the subsurface volume of the wellbore.
 5. The method of claim 4, wherein: the subsurface volume of the wellbore includes casing, rock, and concrete between the casing and the rock; and the defect in the subsurface volume of the wellbore includes a defect in the casing, a defect in the rock, a defect in the concrete, a defect at casing-concrete interface, and/or a defect at rock-concrete interface.
 6. The method of claim 4, wherein an extent of the deviation of the values of the nonlinear material parameters alpha, beta, and delta from the baseline values of the nonlinear material parameters alpha, beta, and delta indicates an extent of the defect in the subsurface volume of the wellbore.
 7. The method of claim 4, wherein a type of the defect in the subsurface volume of the wellbore is determined based on the comparison of the values of the nonlinear material parameters alpha, beta, and delta with the baseline values of the nonlinear material parameters alpha, beta, and delta.
 8. The method of claim 4, wherein a location of the defect in the subsurface volume of the wellbore is determined based on the comparison of the values of the nonlinear material parameters alpha, beta, and delta with the baseline values of the nonlinear material parameters alpha, beta, and delta.
 9. An apparatus that determines wellbore integrity, the apparatus comprising: a first transceiver configured to transmit a pulsed sinusoidal acoustic signal through a subsurface volume of a wellbore, the pulsed sinusoidal acoustic signal having a frequency; and a second transceiver configured to detect the pulsed sinusoidal acoustic signal transmitted though the subsurface volume of the wellbore; wherein: the second transceiver transmits, through the subsurface volume of the wellbore, a time-reversed acoustic signal of the pulsed sinusoidal acoustic signal, transmission of the time-reversed acoustic signal through the subsurface volume resulting in generation of focused acoustic signal in the subsurface volume of the wellbore; the first transceiver detects the focused acoustic signal in the subsurface volume of the wellbore; and the integrity of the wellbore is determined based on the focused acoustic signal in the subsurface volume of the wellbore, wherein determination of the integrity of the wellbore based on the focused acoustic signal in the subsurface volume of the wellbore includes determination of values of nonlinear material parameters alpha, beta, and delta in the subsurface volume of the wellbore based on the focused acoustic signal, and comparison of the values of the nonlinear material parameters alpha, beta, and delta with baseline values of the nonlinear material parameters alpha, beta, and delta, wherein the baseline values of the nonlinear material parameters alpha, beta, and delta are previously measured values of the nonlinear material parameters alpha, beta, and delta with known integrity of the wellbore, further wherein the nonlinear material parameter alpha relates to a strength of hysteresis and the nonlinear material parameters beta and delta relate to acoustoelasticity.
 10. The system of claim 9, wherein the frequency of the pulsed sinusoidal acoustic signal determines a size of the subsurface volume of the wellbore that is probed, and the integrity of the wellbore is determined by increasing the size of the subsurface volume of the wellbore that is probed via one or more changes in the frequency of the pulsed sinusoidal acoustic signal, further wherein characteristics of subsurface materials determined by an initial probing of the subsurface volume of the wellbore with a smaller size of the probing is accounted for in a later probing of the subsurface volume of the wellbore with a larger size of the probing to determine characteristics of subsurface materials included in the larger size of the probing and not included in the smaller size of the probing.
 11. The system of claim 9, wherein the nonlinear material parameters alpha, beta, and delta include higher-order elastic moduli.
 12. The system of claim 9, where deviation of the values of the nonlinear material parameters alpha, beta, and delta from the baseline values of the nonlinear material parameters alpha, beta, and delta indicates formation of a defect in the subsurface volume of the wellbore.
 13. The system of claim 12, wherein: the subsurface volume of the wellbore includes casing, rock, and concrete between the casing and the rock; and the defect in the subsurface volume of the wellbore includes a defect in the casing, a defect in the rock, a defect in the concrete, a defect at casing-concrete interface, and/or a defect at rock-concrete interface.
 14. The system of claim 12, wherein an extent of the deviation of the values of the nonlinear material parameters alpha, beta, and delta from the baseline values of the nonlinear material parameters alpha, beta, and delta indicates an extent of the defect in the subsurface volume of the wellbore.
 15. The system of claim 12, wherein a type of the defect in the subsurface volume of the wellbore is determined based on the comparison of the values of the nonlinear material parameters alpha, beta, and delta with the baseline values of the nonlinear material parameters alpha, beta, and delta.
 16. The system of claim 12, wherein a location of the defect in the subsurface volume of the wellbore is determined based on the comparison of the values of the nonlinear material parameters alpha, beta, and delta with the baseline values of the nonlinear material parameters alpha, beta, and delta. 